We report numerical simulations of surface modes in two-dimensional high-index-contrast photonic crystal slabs with a flat dielectric margin of width on the order of the photonic crystal periodicity. Our calculations using plane wave expansion method reveal multiple surface guided modes within the photonic band gap, with some high-order modes exhibit relatively flat dispersion curves. We calculate the finite-length surface waveguide modes transmission and field patterns using two-dimensional finite-difference time-domain method. We verify the surface mode dispersion curves by using spatial Fourier transform of the mode field patterns. Our study on surface modes under small ambient refractive index changes (5 × 10-3) shows that lower order modes exhibit larger wavelength shifts on the order of 1 nm. We also design a 4-port 3-channel bidirectional coupler using a conventional dielectric waveguide side coupled to the multimode surface waveguide.
© 2006 Optical Society of America
Surface waves at the interface between a semi-infinite periodic layered dielectric medium and a homogeneous medium has long been studied as electromagnetic Bloch modes . In general, at any terminations of a two-dimensional (2-D) or three-dimensional photonic crystal (PC), surface modes exist as localized electromagnetic field distributions [2–5]. Various surface wave related phenomena have also been explored including enhanced transmission through slits in PC slabs and strong beaming of light emerging from a PC waveguide , and leaky surface modes assisted transmissions in the forbidden band . Truncated 2-D PC slabs also support surface waves, as calculated and experimentally observed in a double-trench PC defect waveguide [8–10]. These surface waves are localized at the interface between the trench and the PC region, rather than guided in the defect waveguide region. It has also been recognized that at frequencies near the PC Brillouin zone edge, surface waves exhibit a relatively large group refractive index (implying a slow group velocity), and thus a surface-mode microcavity is expected to be high-Q [11, 12].
Here we report numerical simulations of surface waves guided in a relatively wide and flat dielectric margin in a high-index-contrast 2-D PC slab [Fig. 1(a)]. Lightwave in such surface waveguide is partially confined by total internal reflections at the flat sidewall and by coherent scatterings at the PC holes lattice. The PC slab gives rise to much of the interesting dispersion characteristics similar to those of line defect waveguides in PC slabs. Compared to truncated PC slabs that also support surface states, a wide margin in structure under study allows fabrication tolerance, and also introduces multiple partially transmission modes in the band gap that can be relevant for multi-channel wavelength multiplexing applications (e.g. in passive optical networks (PON)). The flat sidewall enables convenient lateral evanescent coupling to the surface modes, e.g. through coupling the finite-size PC slab to a conventional dielectric waveguide [Fig. 1(c)]. The coupling forms an asymmetric coupler with regular waveguide modes coupled to highly dispersive surface modes. Another example is our previously proposed surface guided photonic crystal embedded microcavities (PCEMs) [13, 14]. The surface PC waveguide can be configured to form a closed loop in a hexagonal-shaped microresonator. A conventional strip waveguide can then be side-coupled to one surface waveguide facet of such hexagonal microcavity that has embedded PC holes array.
2. Plane wave expansion calculations
Figure 1 shows the schematic of a 2-D PC slab surface waveguide. The PC slab comprises a triangular lattice of air holes, with lattice constant a and hole radius r. The surface waveguide axis is in the z-direction (ΓK symmetry direction). The waveguide transverse dimension is in the x-direction (ΓM symmetry direction). The dielectric margin of width m spanning between the flat sidewall and the edge of the first row of holes constitutes an asymmetric waveguide. We focus on the E-modes (electric field parallel to plane of the periodicity) in such PC slab surface waveguides.
We first apply the plane wave expansion (PWE) method with super-cell approach in order to numerically calculate the projected dispersion diagrams of the PC slab surface waveguides of an infinite length. It is important to note that for r = 0.3a and refractive index n = 3.5 (e.g. silicon refractive index), the PC slab displays a band gap between ~0.205 c/a and ~0.275 c/a, where c is the speed of light in vacuum. We consider surface waveguides for a range of margin widths from m = 0.1a to m = 2.0a. We define a super-cell as one unit cell in the z-direction (ΓK) and 11 unit cells in the x-direction (ΓM), as shown in Fig. 1(b). We discretize each unit cell into 32 computation steps. From the calculated dispersion diagrams, we extract several pertinent parameters about the surface waveguide modes including the group refractive index n g, the guided mode bandwidth, and the effective span in projected wavevector k that displays a relatively large n g.
Figures 2(a)–2(c) show the representative projected dispersion diagrams (projected to the z- direction) for surface waveguides of m = 0.1a, 0.7a, and 1.2a. The band gap can be clearly discerned. The thin dark lines are the PC slab modes in the 1st and 2nd PC bands. The dashed line is the light line. We highlight the surface modes and label them as A0, A1, etc. For m = 0.1a [Fig. 2(a)], we observe only a single surface mode A0. For m = 0.7a [Fig. 2(b)], two surface modes A1 and A2 exist within the band gap, whereas mode A0 redshifts to below the lower PC zone edge frequency. For m = 1.2a [(Fig. 2(c)], three surface modes A2, A3, and A4 exist within the band gap, whereas modes A0 and A1 redshift to below the lower PC zone edge frequency. It is noteworthy that mode A3 displays a flat band (~0.004c/a bandwidth) over a relatively wide k span (~0.3 (2π/a) to ~0.5 (2π/a)).
Figures 2(d)–2(g) show the PWE-calculated field patterns at the zone edge frequencies for modes A0 – A3 with various m’s. Modes A0 and A1 exhibit familiar field patterns as previously shown in corrugated surface waveguides . Whereas, modes A2 and A3 display two field extrema in the x-direction.
Figure 2(h) shows the PWE-calculated surface modes zone edge frequencies for m spanning from 0.1a to 2.0a. We identify a total of six different guided modes. As m increases, the surface modes dispersion curves and thereby the zone edge frequencies redshift, and the number of surface modes within the band gap increases. The surface waveguide exhibits single guided mode (fundamental mode) in the PC band gap only for m below ~0.3a.
Figure 3(a) shows the surface modes k span (Δk) within which the modes exhibit a relatively large n g (arbitrarily set at exceeding n g = 50) as a function of the zone-edge frequency. We find that Δk overall narrows as the mode redshifts with the margin width. For example, mode A2 displays a Δk that is narrowed from ~0.06 (2π/a) to ~0.01 (2π/a) with m widens from 0.6a to 1.4a. Mode A2 exhibits ~3 to ~5 times wider Δk than mode A1 at the same zone-edge frequencies. Whereas mode A4 only displays marginally wider Δk than mode A1. It is noteworthy that mode A3 Δk exceeds ~0.1 (2π/a) for a relatively wide span of m between 1.1a and 1.7a [see Fig. 2(h)]. We remark that the lowest order mode A0 also exhibits wide Δk up to 0.1 (2π/a) in m = 0.1a waveguide, which is however prohibitively narrow and not favorable for device fabrication. Increasing the margin width to 0.2a renders A0 mode Δk narrows to 0.01 (2π/a). Figures 3(b)–3(d) show the PWE-calculated dispersion diagrams for m = 1.1a, m = 1.7a, and m = 1.9a, with mode A3 Δk of ~0.128 (2π/a), ~0.112 (2π/a), and 0.054 (2π/a), respectively.
3. Finite-difference time-domain calculations
Here we examine surface waveguides of finite lengths L = 16a and L = 32a by means of 2-D finite-difference time-domain (FDTD) simulations. We employ 36 computation points per period in our FDTD calculations. We launch at longitudinal position z = 0 a single-pulse source (pulse width c Δt ~ 4.3a) with center frequency ~0.2322 c/a, and detect the transmission field at the waveguide forward end and the reflection field at the waveguide backward end. We apply perfectly matched layers (PMLs) of reflectivity 10-8 and thickness of 0.5 μm to all four boundaries of the computation domain [Fig. 1(a)]. The finite simulation domain effectively imposes mode-mismatch reflections at the two waveguide ends.
Figure 4(b) shows the simulated transmission spectra for m = 0.8a with L = 16a and L = 32a. The two waveguides show largely comparable spectra with sharper spectral features in the case of L = 32a. In both cases we observe two transmission bands. We contrast the transmission spectra with the PWE-calculated dispersion diagram, as shown in Fig. 4(a). We find that the two transmission bands lie in the band gap. The minor one corresponds to mode A1, with peak transmissions below ~0.3 and spans from ~0.2 c/a to ~0.209 c/a. The transmission bandwidth is bound by mode A1 zone-edge frequency and the maximum frequency of the mode A1 dispersion curve. The major one corresponds to mode A2, with peak transmissions of ~0.75 and spans from ~0.233 c/a to ~0.26 c/a. The transmission bandwidth is bound by mode A2 zone-edge frequency and the intersection between the light line and the mode A2 dispersion curve. In between the two transmission bands is a forbidden band with a bandwidth of ~0.024 c/a.
Figure 4(c) shows the corresponding simulated reflection spectra. We see strong reflections within the forbidden band. Reflection corresponding to mode A1 is relatively high (below ~0.5), whereas reflection corresponding to mode A2 is relatively low (below ~0.25). We attribute the reflection to the mode-mismatch reflection at the forward end of the surface waveguide. However, we note that for each waveguide the transmission and reflection spectra are not entirely complimentary, thus suggesting that light energy is partially lost to radiation modes in the PC slab and in the air.
Figures 4(d) and 4(e) show the zoom-in view of the transmission bands of modes A1 and A2. The spectra show strong oscillations, with a smaller period shown in the L = 32a spectrum than in the L = 16a spectrum. We attribute the spectral oscillations to the Fabry-Perot (FP) reflections between the waveguide end-faces (mode-mismatch reflections), similar to the experimentally observed FP resonance spectra in line defect waveguides .
Figures 5(a) and 5(b) show two representative FDTD-simulated time-averaged steady-state H-field intensity patterns (L = 32a, m = 0.8a) at (a) mode A1 transmission peak at 0.20016 c/a (near the zone-edge frequency) and (b) mode A2 transmission peak at 0.23314 c/a (near the zone-edge frequency). Figures 5(c) and 5(d) show the zoom-in view of the steady-state H-field amplitude patterns.
We apply spatial Fourier transform to the simulated field patterns in order to extract k-space information of the surface modes. Figure 5(e) shows the Fourier transform of mode A1 field pattern [Fig. 5(c)]. We identify three peaks in the k-space, with major k z component at ~0.5098 (2π/a), and minor kz components at ~0.246 (2π/a) and ~0.754 (2π/a). We fold kz components exceeding 0.5 (2π/a) back into the 1st Brillouin zone (BZ). The three kz components are then reduced to two projected wavevectors (marked with *’s) at ~0.4902 (2π/a) and ~0.246 (2π/a), as shown in Fig. 4(a).
Figure 5(f) shows the Fourier transform of mode A2 field pattern [Fig. 5(d)]. We identify only one peak in the k-space at kz ~ 0.50098 (2π/a), corresponding to kz ~ 0.49902 (2π/a) in the 1st BZ. Other retrieved wavevector values in Fig. 4(a) are obtained in the same fashion from the field patterns at various frequencies using L = 32a. The Fourier-transform retrieved k-vectors show excellent agreement with the PWE-calculated surface mode dispersion curves.
We also examine the higher order modes A3 and A4 with m = 1.5a waveguide. Figure 6(a) shows the PWE-calculated projected dispersion diagram. The Fourier-transform retrieved k-vectors are shown as symbols. Figures 6(b) and 6(c) show the FDTD-simulated H-field patterns for modes A3 at 0.2185 c/a and A4 at 0.2391 c/a. Figure 6(d) shows the k-space distribution of the simulated mode A3 field pattern [Fig. 6(c)]. We observe five peaks in the k-space distribution, corresponding to three wavevectors (marked with *’s) on mode A3 dispersion curve in Fig. 6(a). We see that the retrieved k-vectors confirm the flat band characteristics of mode A3.
Figure 6(e) shows the Fourier analysis of mode A4 field pattern. Three major k-space peaks are folded to two wavevectors on mode A4 dispersion curve in Fig. 6(a). The retrieved k-vectors to the left of the light line are due to the k-vectors in the 2nd BZ folded back to the 1st BZ.
4. Ambient refractive index sensing
Here we examine effects of small refractive index changes on the surface wave transmission. We tune the refractive index of the ambient cladding and the PC holes lattice, with an eye on potential applications in microfluidic sensing [16–18]. We consider the refractive index change in water (e.g. due to a presence of targeted biochemical molecules). We vary the ambient refractive index from n c = 1.33, with Δnc= 0.005. We assume surface waveguide of periodicity a = 0.36 μm and L = 32a. We use the Fabry-Perot spectra to identify the wavelength shifts.
Figure 7(a) shows the transmission spectra around mode A1 zone-edge frequency with two ambient refractive indices. Mode A1 spectra redshifts by ~1.3 nm as the ambient refractive index rises from 1.33 to 1.335. It is worth mentioning that the FP resonance wavelengths redshifts and the zone-edge wavelength redshift with such an ambient refractive change are comparable to the reported resonance wavelength shifts in a PC-based microcavity .
Figure 7(b) shows the Δnc induced fractional frequency shifts |Δ(c/a)| / (c/a) as a function of center frequency (at selected transmission peaks) for modes A1, A2, and A3. The fractional frequency shift increases with the center frequency (with narrowing margin widths). Mode A1 exhibits largest fractional frequency shifts (~1.0×10-3) to ambient refractive index variations among the three modes. Figures 7(c)–7(e) show the time-averaged E-field intensity distribution for modes A1, A2 and A3. The E-field intensity filling factor in the ambient region (integrated E-field intensity normalized by the computation area as shown) for mode A1 is ~58.1%, and for modes A2 and A3 are ~51.1% and ~50.6%, respectively.
5. Asymmetric 4-port 3-channel bi-directional coupler design
In order to further illustrate potential functionalities of multimode surface waveguides, we show an initial design of a surface waveguide side-coupled to a conventional strip waveguide as a 4-port multi-wavelengths bi-directional coupler [see schematic in Fig. 1(c)]. The design is motivated by potential passive optical network (PON) applications, with multiplexed wavelength channels of about 1.31 μm, 1.49 μm, and 1.55 μm. We assume silicon photonic devices with a device refractive index of 3.5 and air in the cladding and PC holes. We adopt the PC periodicity a = 0.33 μm, margin of 0.23 μm (m ~ 0.7a), and a strip waveguide of w = 0.25 μm. The lateral interaction length is 15.18 μm (L = 46a) and the coupling air gap distance is 0.2 μm.
Figure 8(a) shows the FDTD-simulated spectra of output ports 2–4. At throughput port 2, the spectrum shows an overall wide-band transmission except for two pronounced transmission dips around 1.31 μm (3-dB bandwidth ~ 10 nm) and 1.55 μm (3-dB bandwidth ~ 26 nm). The forward coupling (port 3) shows a single transmission peak around 1.31 μm, whereas the backward coupling (port 4) shows a single reflection peak around 1.55 μm.
Figure 8(b) shows the PWE-calculated dispersion diagram of surface modes A1 and A2, and of the strip waveguide. The dispersion lines cross at (k1 ω 1) ≈ (0.4406 (2π/a), 0.2138 c/d) and (k2, ω 2) ≈ (0.3775 (2π/a), 0.2516 c/a), suggesting phase matching between the surface modes and the strip waveguide mode. The transmission dips wavelengths correspond well to the two intersections normalized frequencies.
Figures 8(c)–8(e) show the FDTD-simulated steady-state H-field intensity patterns at wavelengths 1.3114 μm (forward coupling), 1.49 μm (throughput transmission), and 1.5494 μm (backward coupling). It is noteworthy that the waveguide-coupled surface mode field intensities suggest field enhancement.
We apply spatial Fourier transform analysis to the coupling regions (denoted by dashed-line windows) in order to extract k-space information of the forward and backward coupling modes. Figures 8(f) and 8(g) show the Fourier-transformed k-space representations at λ = 1.3114 μm and λ= 1.5494 μm. For the forward coupling mode, we only discern one major k-space component in the 2nd BZ with kz ~ 0.624 (2π/a), representing co-propagating phase-matched strip waveguide mode and mode A2. We remark that the kz value corresponds to ~0.376 (2π/a) in the 1st BZ. For the backward coupling mode, we discern one major k-space component in the 1st BZ with kz ~ 0.445 (2π/a), representing forward propagating strip waveguide mode, and another one in the 2nd BZ in the negative domain with kz ~ -0.553 (2π/a), representing backward propagating mode A1. We note that the kz value in the 2nd BZ corresponds to ~0.447 (2π/a) in the 1st BZ.
We explain the bi-directional two-mode coupling using a schematic extended band diagram spanning the full 1st and part of 2nd BZ’s, as shown in Fig. 8(h). According to the Fourier-transformed k-space, modes A1 and A2 dispersion lines are in the 2nd BZ, whereas the waveguide dispersion line extends from the 1st to the 2nd BZ. We see that the forward propagating waveguide mode at ω 2 phase matches with mode A2 (kWG2 = kA2), suggesting directional coupling. In contrast, the forward propagating waveguide mode at ω 1 only phase matches with mode A1 with an additional momentum of 2π/a (kWG1 = 2π/a + kA1), suggesting Bragg-reflection assisted counter-directional coupling.
In conclusion, we have numerically studied surface modes in a two-dimensional photonic crystal surface waveguide with a flat dielectric margin and a finite length. We examined the multiple guided modes inside the photonic band gap as a function of the margin width spanning from 0.1a to 2.0a. Our numerical analysis based on plane wave expansion method suggests that some high-order modes demonstrate group refractive indices exceeding 50 with a relatively wide k-span exceeding 0.1 (2π/a). We used two-dimensional FDTD to evaluate the finite-length surface guided mode transmission and reflection and their mode-field patterns. We apply spatial Fourier analysis on the simulated mode-field patterns and verify the flat bands in surface wave dispersion curves. For ambient index variation of Δnc/nc = 0.005/1.33, wavelength redshifts of |Δ(c/a)| / (c/a) ~ 1×10-3 can be attained.
The surface waveguide provides one interface for side-coupling to other photonic components or provides a useful platform for more sophisticated device architectures. We demonstrated a simple example with the surface waveguide laterally coupled to a conventional strip waveguide. Such coupling forms a 4-port bi-directional coupler with potential to realize three wavelengths multiplexer/demultiplexer (e.g. for passive optical networks (PON) applications). We envision that the surface waveguide can be incorporated with other photonic devices in order to realize further functionalities such as dispersion control, buffering, and (biochemical) sensing.
The work described in this paper was substantially supported by grants from the Research Grants Council and from the University Grants Committee of the Hong Kong Special Administrative Region, China. H. Chen’s studentship was supported by grant HKUST6112/03E.
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